Download Mathematical model of Coriolis effect

Please cite as: Inż. Ap. Chem. 2017, 56, 1, 23-25
Nr 1/2017
INŻYNIERIA I APARATURA CHEMICZNA
str. 23
Edward SOBCZAK1, Jerzy KASPRZAK2, Tomasz RINGEL1
e-mail: [email protected]
1
2
Wydział Technologii i Inżynierii Chemicznej, Uniwersytet Technologiczno-Przyrodniczy, Bydgoszcz
Wojewódzka Stacja Sanitarno-Epidemiologiczna, Bydgoszcz
Mathematical model of Coriolis effect
Introduction
The Coriolis effect occurs in the rotating reference systems. For an
observer remaining in such a system it manifests as deflection of a path
of an object moving within a rotating system. This deflection seems to
be generated by a force, the so-called Coriolis force. The Coriolis force
is an apparent effect that occurs only in rotating non-inertial systems. For
an external observer such force does not exist. For him the system
changes its location while a moving body maintains its motional state in
agreement with the first principle of dynamics.
A French engineer and mathematician Gaspard Gustave Coriolis was
an inventor of this effect whilst its first experimental confirmation for the
Earth has been demonstrated by Jean Bernard Léon Foucault who used
his pendulum [Bogusz et al., 2010; Skorko, 1982].
The Foucault pendulum [Bogusz et al., 2010] that was used to prove the
existence of the Coriolis force, changes its plane of oscillations influenced by the effect of the horizontal component of the Coriolis force
(Fc), resulting in its rotation in an opposite direction with respect to the
Earth rotation while at the same time the pendulum weight draws
a rosette. Under the action of the Coriolis force the plane of oscillations
of the mathematical pendulum z changes with the angular velocity
ω = ωz sinϕ with a period
T
T= z
(1)
sin ϕ
The rotation of the plane of oscillations is the fastest at the poles
(T = 24 h), and disappears at the equator (T = ∞). In Poland the
period T equals from 30 to 32 hours.
Treating the pendulum motion [Bogusz et al., 2010] of length
l = 28 m at small deflections as harmonic, the equations of motion
with accounting components of the Coriolis force are as follows
m
d2x
dt 2
= − kx + 2mω z
dy
dt
d2 y
dx
= − ky − 2mω z
dt 2
dt
The Coriolis force is given as
m
Fc = 2 m (ω × ν )
(2)
(3)
(4)
while the acceleration involved with this force is as follows
(5)
ac = −2 (ω × ν )
where:
m – body mass,
ν – body velocity
ω – angular velocity of the system.
A simple proof on the Coriolis acceleration can be offered
[Skorko, 1982] a distance passed by a sphere with linear velocity ν
along the radius of the spinning disc is equal to ∆r = ν t, while
a motionless observer will state that the path made along an arc of
the spinning disc during its rotation by an angle: φ = ω t will be
1 2
ac t
2
Hence the Coriolis acceleration, ac, of a sphere is equal to
(6)
ac = 2ων
(7)
∆S = ϕ∆r = (ωt )vt =
An example of the existence of the Coriolis acceleration for
a falling sphere onto a spinning Earth is discussed in [Bogusz et al.,
2010]. To determine the acceleration
a=
d 2r 1
= ( F + Fod + Fc ) = g − ω 2 r − 2ων
dt 2 m
(8)
the gravity force, F = mg , the centrifugal force Fcen = mω 2 r and the
Coriolis force, Fc = 2mων . The effect of the Coriolis force on
spinning single stars and a system of binary stars has been derived
by [Lal et al., 2008; 2009] of the Coriolis force on thermal convection and impurity segregation during crystal growth under
∂ν
1
+ ν∇ν = − ∇P +ν∇ ⋅ (ν∇F + ∇ν ) − 2(ω ×ν ) + βg (T − Tm ) (9)
∂t
ρ
microgravity [Müller et al., 1992; Kohno i Tanahashi, 2002].
Investigations of the effect of the Coriolis force on hydrodynamics of the liquid flow have been carried out by [Martinez et al.,
2012] and on erosion of impellers and spinning cylindrical mixers –
by Krupicz [2000] and Gałkowski et al., [1999] and on the separation
effectiveness of suspensions [Łagutkin, 2003], Choi and Yoo [2015]
utilizing the Coriolis effect fCoriolis = −2ωvρ and the different values
of the ρ density of these particles the separation for DNA
purification.
Investigations of the effect Coriolis on the astrophysics problems
shapes of rotating stars or secondary circulation around headlands
and islands and the problem of the physical nature of the EastAntarctic atmospheric boundary layer and regional climate model
and other the problem meteorology results of a it by using the following relationship (Eq. 4), which does not agree with definition of
acceleration of an object moving with a velocity v = dr / dt along
a radius of a spinning disk with the angular velocity ω
ac =
dvr d (ωr )
dr
=
=ω
= ων
dt
dt
dt
(10)
Investigations of the effect of the Coriolis force on hydrodynamics of the liquid flow and on the thermal convection and segregation
during crystal growth and on the separation effectiveness of suspensions and on the separation for DNA purification and other
chemical engineering and bioengineering process results by
using the Eqs (4) and (5).
New mathematical model of Coriolis effect
The object that moves along a meridian of the Earth globe with
a linear velocity v covers a distance dt during the time ds and
changes its location (geographical latitude) with respect to the equadS
tor plane by an angle dϕ =
and a distance r = R cos ϕ measR
ured from the axis of the Earth rotation and thus changes linear
2πr
spinning velocity around the axis vr =
= ωr = ωR cos ϕ , hence
T
v=
dS Rdϕ
=
dt
dt
ϕ ∈ (−
π π
, )
2 2
(11)
Please cite as: Inż. Ap. Chem. 2017, 56, 1, 23-25
str. 24
INŻYNIERIA I APARATURA CHEMICZNA
Making use of the definition of acceleration (tangential to a latitude of the radius r)
ac =
dυ r
d (ωR cos ϕ )
dϕ
=
= ωR (− sin ϕ )
dt
dt
dt
(12)
and the object linear velocity along a meridian (Eq. 11) one will get
an expression for the Coriolis acceleration
or
ac = ω (− sin ϕ )v
(13)
ac = − (ω × ν )
(14)
ac = Rω 2 sinϕ (1 +
Nr 1/2017
s
)s
cos ϕ
s=
vs
ωR
From the graphs drawn for a dependence of the Coriolis
acceleration on the reduced object velocity, s, (for
v = ±v S = ±s ω R ) and increase in the absolute value of
the Coriolis acceleration at vS > 0 has been observed while at
v S < 0 a similar decrease in the absolute value obtained
(Eqs (15) or (18) and Figs. 1÷4.
In agreement with the Newton’s inertia principle ( Fc = − m ac ) an
equation for the Coriolis force takes the form
Fc = m(ω × v)
(15)
If the object at the same time moves along a meridian
with a velocity v and along a parallel of latitude with a velocity
v s = sωR , assuming s ∈ (−1,+1) , its angular velocity of
circulation around the Earth globe will change by ∆ω =
υs
,
r
therefore the expression for the Coriolis acceleration (13)
will take the following form
a c = (ω +
vs

s 

)(− sin ϕ )v = −(ων sin ϕ )1 +
r
 cosϕ 
(16)
Fig. 1. Dependence of Coriolis acceleration on the velocity v of object
motion simultaneously along meridian within the range φ = (-75º, +75º)
and along the parallel of latitude at velocity vS (|v| = |vS|, s = vS/ωR) for
(-) ← vS and v↓ (-)} and for {vS → (+) and v ↑ (+)}
or
ac = −(ω × v)(1 +
s ∈ (−1,+1)
v
s
) s= s
cos ϕ
ωR
2π
ω=
,
T
(17)
and according to the Newton’s inertia principle ( F = − m a ) the
Coriolis force will be

s 

Fc = m(ω × v)1 +
 cos ϕ 
(18)
After substituting the positive components {vs , v} > 0 directed east(+ )
wards vs → (+ ) and northwards v ↑ as well as the negative
{vs , v} < 0 directed westwards (− ) ← vs and southwards v ↓
(− )
Fig. 2 Dependence of Coriolis acceleration on the velocity v of object
motion simultanousy along meridian within the range φ = (-75º, +75º)
and along the parallel of latitude at velocity vS (|v| = |vS|, s = vS/ωR) for
{ vS → (-) and v↑ (+)} and for {vS → (+) and v↑ (+)}
(according to a scheme (− ) ← vs → (+ ) and v b+− ) with a reduced
object velocity along the parallel of latitude s = vs , s ∈ (−1,+1)
ωR
v
as well as meridian k s =
, k s ∈ (−1,+1) , for k s = ± s a modified
Rω
form of Eq. (16) has been obtained
ac = − Rω 2 sin ϕ (1 +
s
)k s
cos ϕ
(19)
for v = vs → v = ± v s and sign(v)=sign(vs) (ks=s)
ac = − Rω 2 sin ϕ (1 +
s
)s
cos ϕ
for sign(v) = -sign(vs) (ks=-s)
(20)
(21)
Fig. 3 Dependence of Coriolis acceleration on the velocity v of object
motion simultanouly along meridian within the range φ = (-75º, +75º)
and along the parallel of latitude at velocity vS (|v| = |vS|, s = vS/ωR) for
{(-) ← vS and v↑ (+)} and for {vS → (+) and v↓ (-)}
Please cite as: Inż. Ap. Chem. 2017, 56, 1, 23-25
Nr 1/2017
INŻYNIERIA I APARATURA CHEMICZNA
str. 25
In the case launching rocket from the point at latitude φ0 > 0
with a velocity (+vs,+ v), in the result of the Coriolis effect track
deviation bullet from the meridian (∆S s > 0) in the east direction
has been obtained:
∆S s = vs ∆t + ωv(sin ϕ s )(∆t ) 2
(22)
where
∆t =
R(ϕ − ϕo )
v
1
2
ϕ s = (ϕ + ϕ o )
LITERATURA
Fig. 4 Dependence of Coriolis acceleration on the velocity v of object
motion simultanouly along meridian within the range φ = (-75º, +75º)
and along the parallel of latitude at velocity vS (|v| = |vS|, s = vS/ωR) for
{(-) ← vS and v↓ (-)} and for {vS → (+) and v↑ (-)}
Bogusz W., Garbarczyk J., Krok F., (2010).
(in Polish). PWN, Warszawa, Chapter 9
Fundamentals of physics
Choi S., Yoo J-C., (2015). Automated centrifugal-microfluidic platform for
dna purification using laser burst valve and Coriolis effect. Appl. Biochem. Biotechnol., 175:3778–3787 DOI 10.1007/s12010-015-1546-x6
Gałkowski Z., Pietrzakowski M., Tylikowski A., (1999). Dynamics of the
closed isotropic shell rotating at a constant angular speed (in Polish)
Pr. Inst. Podstaw Budowy Maszyn Pol. Warszawskiej, 19, 13-25
Discussion
As follows from Eq. (15) modified to the forms (18) and
graphically presented in Figs. 1÷4 for the direction of the component vS → (+) being in accord with that of the Earth surface
vr = ωr, i.e. from west to east, (vS > 0 for s ∈ (0,+1) an increase
of the angular velocity of the Earth globe cycle (ω+ vS/r ) and an
increase of the absolute value of the Coriolis acceleration |a c|.
The derived universal mathematical model (Eq.15) or (Eq.16)
enables calculation of the Coriolis acceleration for a moving object
simultaneously along a meridian or a parallel of latitude (northwards
with a velocity ↑ (v > 0) or southwards with a velocity ↓ (v < 0) and
eastwards with a velocity → (vs > 0) or westwards with a velocity ←
(vs < 0) for geographical latitude φ = (-π/2, π/2).
For a given object velocity along a meridian v ↑ (+) or v ↓ (-)
and a positive velocity along the parallel of latitude vs >0
(eastwards), being in accord with a scheme (-) ← vs → (+),
v (+) ↔ (-) an increase in the angular velocity of the Earth globe
v
cycle by a value of ∆ω = s as well as of an absolute value of the
r
Coriolis force, while for vs < 0 (eastwards) the opposite effect will occur.
Acceleration and Coriolis force will not arise at latitude φ = 0 or for the
velocity component along the meridian v = 0 or for the parallel component:
vs = - ωr = -ωRcosφ directed westwards (Eq. 15).
Kohno H., Tanahashi T., (2002). Finite element simulation of single crystal
growth process using GSMAC method. J. Comp. Appl. Math., 149(1),
359-371. DOI: 10.1016/S0377-0427(02)00543-5
Krupicz B., (2000). Contact stresses in the ventilator blade caused
by Coriolis force (in Polish). Zesz. Nauk., Pol. Białostockiej. Nauki
Techniczne 134, Mechanika 22, 179-187
Lal K., Pathania A., Mohan C., (2008). Effect of Coriolis force on the equilibrium structures of rotating stars and stars in binary systems. Astrophys.
Space Sci. 315, 157-165. DOI: 10.1007/s10509-008-9808-5
Lal K., Pathania A., Mohan C., (2009). Effect of Coriolis force on the shapes
of rotating stars and stars in binary systems. Astrophys. Space Sci. 319,
45-53. DOI: 10.1007/s10509-008-9946-9
Łagutkin G., (2003). Influence of Coriolis force on separation process in
hydrocyclones (in Polish). Czas. Tech., 100, M-5
Martinez J.C., Polatdemir E., Bansal A., Yifeng W., Shengtao W., (2006).
Fluid flow up a spinning egg and the Coriolis force. Eur. J. Physics,
27(4), 805. DOI: 10.1088/0143-0807/27/4/012
Müller G., Neumann G., Weber W., (1992). The growth of homogeneous
semiconductor crystals in a centrifuge by the stabilizing influence of the
Coriolis force. J. Cryst. Growth, 119(1-2), 8–23. DOI: 10.1016/00220248(92)90200-3
Skorko M., (1982). Physics. Manual for students (in Polish). PWN,
Warszawa, 63
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